where \(\dagger \)  indicates the Moore–Penrose pseudoinverse [1]} and \(X=U\Sigma V^\ast \)  is the singular value decomposition.
In many applications, the state dimension \(n\)  is very large, and
forming or storing \(A\)  explicitly becomes impractical.
Instead, we use a rank-\(r\)  approximation for \(A\) , denoted by \(\hat{A}\) , where \(r \ll n\) .
To form \(\hat{A}\) , we construct the optimal rank-\(r\)  approximation for \(X\)  using the truncated singular value decomposition [2]}:
\(X\approx U_r\Sigma _r V_r^\ast \) .
We then project \(A\)  onto the leading \(r\)  principal components of \(X\)  as
\(\hat{A} = U_r^\ast AU_r= U_r^\ast YV_r \Sigma _r^{-1}.\) 
